Non-Gaussianity as a Probe of the Physics of the Primordial Universe

Non-Gaussianity

as a Probe of the Physics of the Primordial Universe

Eiichiro Komatsu

(Texas Cosmology Center, University of Texas at Austin)

Solvary Workshop, “Cosmological Frontiers in Fundamental Physics”

May 13, 2009

1

How Do We Test Inflation?

•

How can we answer a simple question like this:

•

“How were primordial fluctuations generated?”

2

Power Spectrum

•

A very successful explanation (Mukhanov & Chibisov;

Guth & Pi; Hawking; Starobinsky; Bardeen, Steinhardt &

Turner) is:

•

Primordial fluctuations were generated by quantum fluctuations of the scalar field that drove inflation.

•

The prediction: a nearly scale-invariant power spectrum in the curvature perturbation, ζ:

•

^{Pζ(k) = A/k4–ns ~ A/k3}

•

^{where ns}~1 and A is a normalization. ₃

n s <1 Observed

•

The latest results from the WMAP 5-year data:

•

^ns=0.960 ± 0.013 (68%CL; for tensor modes = zero)

•

^ns=0.970 ± 0.015 (68%CL; for tensor modes ≠ zero)

•

tensor-to-scalar ratio < 0.22 (95%CL)

•

^ns≠1: another line of evidence for inflation

•

Detection of non-zero tensor modes is a next important step

Komatsu et al. (2009)

4

Anything Else?

•

One can also look for other signatures of inflation. For example:

•

Isocurvature perturbations

•

Proof of the existence of multiple fields

•

Non-zero spatial curvature

•

Evidence for “Landscape,” if curvature is negative.

Rules out Landscape ideas if positive.

•

Scale-dependent ns (running index)

•

Complex dynamics of inflation ⁵

Anything Else?

•

One can also look for other signatures of inflation. For example:

•

95%CL limits on Isocurvature perturbations

•

^S/(3ζ) <0.089 (axion CDM); <0.021 (curvaton CDM)

•

95%CL limits on Non-zero spatial curvature

•

Ω–1<0.018 (for Ω>1); 1–Ω<0.008 (for Ω<1)

•

95%CL limits on Scale-dependent ns

•

–0.068 < dns/dlnk < 0.012

6

Komatsu et al. (2009)

positive curvature negative curvature

Beyond Power Spectrum

•

All of these are based upon fitting the observed power spectrum.

•

Is there any information one can obtain, beyond the power spectrum?

7

Bispectrum

•

Three-point function!

•

^{Bζ(k1,k2,k3)}

= <ζ_k1ζ_k2ζ_k3> = (amplitude) x (2π)³δ(k₁+k2+k3)b(k1,k2,k3)

8

model-dependent function

k1

k2

k3

Why Study Bispectrum?

•

It probes the interactions of fields - new piece of information that cannot be probed by the power spectrum

•

But, above all, it provides us with a critical test of the simplest models of inflation: “are primordial

fluctuations Gaussian, or non-Gaussian?”

•

Bispectrum vanishes for Gaussian fluctuations.

•

Detection of the bispectrum = detection of non-

Gaussian fluctuations ₁₀

Gaussian?

^WMAP5

11

Take One-point Distribution Function

•The one-point distribution of WMAP map looks pretty Gaussian.

–Left to right: Q (41GHz), V (61GHz), W (94GHz).

•Deviation from Gaussianity is small, if any.

12

Spergel et al. (2008)

Inflation Likes This Result

•

According to inflation (Mukhanov & Chibisov; Guth & Yi;

Hawking; Starobinsky; Bardeen, Steinhardt & Turner), CMB anisotropy was created from quantum

fluctuations of a scalar field in Bunch-Davies vacuum during inflation

•

Successful inflation (with the expansion factor more than e⁶⁰) demands the scalar field be almost interaction-free

•

The wave function of free fields in the ground state is a Gaussian!

13

But, Not Exactly Gaussian

•

Of course, there are always corrections to the simplest statement like this.

•

For one, inflaton field does have interactions. They are simply weak – they are suppressed by the so-called

slow-roll parameter, ε~O(0.01), relative to the free-field action.

14

A Non-linear Correction to Temperature Anisotropy

•

The CMB temperature anisotropy, ΔT/T, is given by the curvature perturbation in the matter-dominated era, Φ.

•

One large scales (the Sachs-Wolfe limit), ΔT/T=–Φ/3.

•

Add a non-linear correction to Φ:

•

^{Φ(x) = Φg(x) + fNL[Φg(x)]2} (Komatsu & Spergel 2001)

•

^fNL was predicted to be small (~0.01) for slow-roll models (Salopek & Bond 1990; Gangui et al. 1994)

15

For the Schwarzschild metric, Φ=+GM/R.

f NL : Form of B ^ζ

•

Φ is related to the primordial curvature perturbation, ζ, as Φ=(3/5)ζ.

•

^{ζ(x) = ζg}(x) + (3/5)fNL[ζg(x)]²

•

^{Bζ(k1,k2,k3)=(6/5)fNL x (2π)3δ(k1+k2+k3) x}

[Pζ(k1)Pζ(k2) + Pζ(k2)Pζ(k3) + Pζ(k3)Pζ(k1)]

16

f NL : Shape of Triangle

•

For a scale-invariant spectrum, Pζ(k)=A/k³,

•

^{Bζ(k1,k2,k3)=(6A2/5)fNL x (2π)3δ(k1+k2+k3)}

x [1/(k1k2)³ + 1/(k2k3)³ + 1/(k3k1)³]

•

Let’s order ki such that k3≤k₂≤k₁. For a given k1, one finds the largest bispectrum when the

smallest k, i.e., k3, is very small.

•

^Bζ(k1,k2,k3) peaks when k3 << k2~k1

•

Therefore, the shape of fNL bispectrum is the squeezed triangle!

(Babich et al. 2004) 17

B ^ζ in the Squeezed Limit

•

In the squeezed limit, the fNL bispectrum becomes:

Bζ(k1,k2,k3) ≈ (12/5)fNL x (2π)³δ(k1+k2+k3) x Pζ(k1)Pζ(k3)

18

Single-field Theorem (Consistency Relation)

•

For ANY single-field models^*, the bispectrum in the squeezed limit is given by

•

^{Bζ(k1,k2,k3) ≈ (1–ns) x (2π)3δ(k1+k2+k3) x Pζ(k1)Pζ(k3)}

•

Therefore, all single-field models predict fNL≈(5/12)(1–n_s).

•

With the current limit ns=0.96, fNL is predicted to be 0.017.

Maldacena (2003); Seery & Lidsey (2005); Creminelli & Zaldarriaga (2004)

* for which the single field is solely responsible for driving inflation and generating observed fluctuations. 19

Understanding the Theorem

•

First, the squeezed triangle correlates one very long-

wavelength mode, kL (=k3), to two shorter wavelength modes, kS (=k1≈k2):

•

^{<ζk1ζk2ζk3}> ≈ <(ζ_kS)²ζ_kL>

•

Then, the question is: “why should (ζ_kS)² ever care about ζ_kL?”

•

The theorem says, “it doesn’t care, if ζ_k is exactly scale invariant.”

20

ζ _kL rescales coordinates

•

The long-wavelength

curvature perturbation rescales the spatial

coordinates (or changes the expansion factor) within a

given Hubble patch:

•

^{ds2=–dt2+[a(t)]2e2ζ(dx)2}

ζ_kL

left the horizon already

Separated by more than H^-1

x1=x0e^ζ1 x2=x0e^ζ2

21

ζ _kL rescales coordinates

•

Now, let’s put small-scale perturbations in.

•

Q. How would the

conformal rescaling of coordinates change the

amplitude of the small-scale perturbation?

ζ_kL

left the horizon already

Separated by more than H^-1

x1=x0e^ζ1 x2=x0e^ζ2 (ζ_kS1)² (ζ_kS2)²

22

ζ _kL rescales coordinates

•

Q. How would the

conformal rescaling of coordinates change the

amplitude of the small-scale perturbation?

•

A. No change, if ζ_k is scale- invariant. In this case, no

correlation between ζ_kL and (ζ_kS)² would arise.

ζ_kL

left the horizon already

Separated by more than H^-1

x1=x0e^ζ1 x2=x0e^ζ2 (ζ_kS1)² (ζ_kS2)²

23

Real-space Proof

•

The 2-point correlation function of short-wavelength modes, ξ=<ζS(x)ζS(y)>, within a given Hubble patch

can be written in terms of its vacuum expectation value (in the absence of ζL), ξ0, as:

•

^{ξζL ≈ ξ0}(|x–y|) + ζL [dξ0(|x–y|)/dζL]

•

^{ξζL ≈ ξ0}(|x–y|) + ζL [dξ0(|x–y|)/dln|x–y|]

•

^{ξζL ≈ ξ0}(|x–y|) + ζL (1–ns)ξ0(|x–y|)

Creminelli & Zaldarriaga (2004); Cheung et al. (2008)

3-pt func. = <(ζS)²ζL> = <ξζLζL>

= (1–ns)ξ0(|x–y|)<ζL2>

•

^ζS(x)

•

^ζS(y)

24

Where was “Single-field”?

•

Where did we assume “single-field” in the proof?

•

For this proof to work, it is crucial that there is only

one dynamical degree of freedom, i.e., it is only ζL that modifies the amplitude of short-wavelength modes, and nothing else modifies it.

•

Also, ζ must be constant outside of the horizon

(otherwise anything can happen afterwards). This is also the case for single-field inflation models.

25

Therefore...

•

A convincing detection of fNL > 1 would rule out all of the single-field inflation models, regardless of:

•

the form of potential

•

the form of kinetic term (or sound speed)

•

the initial vacuum state

•

A convincing detection of fNL would be a breakthrough.

26

Large Non-Gaussianity from Single-field Inflation

•

^S=(1/2)∫d4x √–g [R–(∂μφ)²–2V(φ)]

•

2nd-order (which gives Pζ)

•

^{S2=∫d4x ε [a3}

⁽

^{∂tζ)2–a(∂iζ)2]}

•

3rd-order (which gives Bζ)

•

^{S3=∫d4x ε2 […a3}

⁽

^{∂tζ)2ζ+…a(∂iζ)2ζ +…a3}

⁽

^{∂tζ)3] + O(ε3)}

27

Cubic-order interactions are suppressed by an additional factor of ε. (Maldacena 2003)

Large Non-Gaussianity from Single-field Inflation

•

^S=(1/2)∫d4x √–g {R–2P[(∂μφ)²,φ]} [general kinetic term]

•

^2nd-order

•

^{S2=∫d4x ε [a3}

⁽

^{∂tζ)2/cs2–a(∂iζ)2]}

•

^3rd-order

•

^{S3=∫d4x ε2 […a3}

⁽

^{∂tζ)2ζ/cs2 +…a(∂iζ)2ζ +…a3}

⁽

^{∂tζ)3/cs2] +}

O(ε³)

28

Some interactions are enhanced for cs2<1.

(Seery & Lidsey 2005; Chen et al. 2007)

“Speed of sound”

cs2=P,X/(P,X+2XP,XX)

•

^S=(1/2)∫d4x √–g {R–2P[(∂μφ)²,φ]} [general kinetic term]

•

^2nd-order

•

^{S2=∫d4x ε [a3}

⁽

^{∂tζ)2/cs2–a(∂iζ)2]}

•

^3rd-order

•

^{S3=∫d4x ε2 […a3}

⁽

^{∂tζ)2ζ/cs2 +…a(∂iζ)2ζ +…a3}

⁽

^{∂tζ)3/cs2] +}

O(ε³)

Large Non-Gaussianity from Single-field Inflation

29

Some interactions are enhanced for cs2<1.

(Seery & Lidsey 2005; Chen et al. 2007)

“Speed of sound”

cs2=P,X/(P,X+2XP,XX)

Another Motivation For f NL

•

In multi-field inflation models, ζ_k can evolve outside the horizon.

•

This evolution can give rise to non-Gaussianity;

however, causality demands that the form of non-

Gaussianity must be local!

Separated by more than H^-1

x1 x2

ζ(x)=ζg(x)+(3/5)fNL[ζg(x)]²+Aχg(x)+B[χg(x)]²+… 30

Now:

•

I hope that I could convince you that fNL is a very powerful quantity for testing single-field inflation models.

•

Let’s look at the observational data!

31

32

Decoding Bispectrum

•

Hydrodynamics at z=1090 generates acoustic

oscillations in the bispectrum

•

Well understood at the linear level (Komatsu &

Spergel 2001)

•

Non-linear extension?

•

Nitta, Komatsu, Bartolo,

Matarrese & Riotto, arXiv:

0903.0894

•

^fNLlocal~0.5

Measurement

•

Use everybody’s favorite: χ² minimization.

•

^Minimize:

•

with respect to Ai=(fNLlocal, fNLequilateral, bsrc)

•

^Bobs is the observed bispectrum

•

^B(i) is the theoretical template from various predictions

33

Journal on f NL (95%CL)

•

–3500 < fNL < 2000 [COBE 4yr, lmax=20 ]

•

^{–58 < fNL} < 134 [WMAP 1yr, lmax=265]

•

^{–54 < fNL} < 114 [WMAP 3yr, lmax=350]

•

^{–9 < fNLlocal} < 111 [WMAP 5yr, lmax=500]

Komatsu et al. (2002) Komatsu et al. (2003)

Spergel et al. (2007) Komatsu et al. (2008)

34

Latest on f NL

•

CMB (WMAP5 + most optimal bispectrum estimator)

•

^{–4 < fNL} < 80 (95%CL)

•

^fNL = 38 ± 21 (68%CL)

•

Large-scale Structure (Using the SDSS power spectra)

•

^{–29 < fNL} < 70 (95%CL)

•

^{fNL = 31 +16–27 (68%CL)}

Smith, Senatore & Zaldarriaga (2009)

Slosar et al. (2009)

(Fast-moving field!)

35

Weak 2- σ “Hint”?

•

So, currently we have something like fNL~40±20 from the WMAP 5-year data, and 30±15 from WMAP5+LSS.

•

Without a doubt, we need more data...

•

WMAP7 is coming up (early next year)

•

WMAP9 in ~2011–2012

•

^And...

36

Planck!

•

Planck satellite is scheduled to be launched TOMORROW, from French Guiana.

•

Planck’s expected 68%CL errorbar is ~5.

•

Therefore, if fNL~40, we would see it at 8σ. If ~30, 6σ.

Either way, IF (big if) fNL~30–40, we will see it

unambiguously with Planck, which is expected to deliver the first-year results in ≥2012.

37

Trispectrum: Next Frontier?

•

The local form bispectrum,

Βζ(k1,k2,k3)^=(2π)3δ(k1+k2+k3)fNL[(6/5)Pζ(k1)Pζ(k2)+cyc.]

•

is equivalent to having the curvature perturbation in position space, in the form of:

•

^ζ(x)=ζg(x) + (3/5)fNL[ζg(x)]²

•

This provides a useful model to parametrize non-

Gaussianity, and generate initial conditions for, e.g., N-body simulations.

•

This can be extended to higher-order:

•

^ζ(x)=ζg(x) + (3/5)fNL[ζg(x)]² + (9/25)gNL[ζg(x)]^{3 38}

Local Form Trispectrum

•

^{For ζ(x)=ζg}(x) + (3/5)fNL[ζg(x)]² + (9/25)gNL[ζg(x)]³, we obtain the trispectrum:

•

^{Tζ(k1,k2,k3,k4)=(2π)3δ(k1+k2+k3+k4)}

{gNL[(54/25)Pζ(k1)Pζ(k2)Pζ(k3)+cyc.] +

(fNL)²[(18/25)Pζ(k1)Pζ(k2)(Pζ(|k1+k3|)+Pζ(|k1+k4|))+cyc.]}

k3

k4

k2

k1

g NL

k2

k1

k3

k4

f NL 2

³⁹

Trispectrum: if f

^NL

is ~50,

excellent cross-check for Planck

• Trispectrum (~

^f_NL²

)

• Bispectrum (~

^f_NL

)

Kogo & Komatsu (2006)

40

(Slightly) Generalized Trispectrum

•

^{Tζ(k1,k2,k3,k4)=(2π)3δ(k1+k2+k3+k4)}

{gNL[(54/25)Pζ(k1)Pζ(k2)Pζ(k3)+cyc.]

+τ_NL[(18/25)Pζ(k1)Pζ(k2)(Pζ(|k1+k3|)+Pζ(|k1+k4|))+cyc.]}

The local form consistency relation, τ_NL=(fNL)², may not be respected – additional test of multi-field inflation!

k3

k4

k2

k1

g NL

k2

k1

k3

k4

f NL 2

⁴¹

Trispectrum: Next Frontier

•

A new phenomenon: many talks given at the IPMU non- Gaussianity workshop emphasized the importance of

the trispectrum as a source of additional information on the physics of inflation.

•

^{τNL ~ fNL2; τNL ~ fNL4/3; τNL} ~ (isocurv.)*fNL2; gNL ~ fNL; gNL ~ fNL2; or they are completely independent

•

Shape dependence? (Squares from ghost condensate, diamonds and rectangles from multi-field, etc)

42

Large-scale Structure of the Universe

•

New frontier: large-scale structure of the universe as a probe of primordial non-Gaussianity

43

New, Powerful Probe of f

NL

•f

NL

modifies the power spectrum of galaxies on very large scales

–Dalal et al.; Matarrese & Verde –Mcdonald; Afshordi & Tolley

•The statistical power of this method is VERY promising

–SDSS: –29 < fNL < 70 (95%CL);

Slosar et al.

–Comparable to the WMAP 5-year limit already

–Expected to beat CMB, and reach a

sacred region: fNL~1 ⁴⁴

Effects of f NL on the statistics of PEAKS

•

The effects of fNL on the power spectrum of peaks (i.e., galaxies) are profound.

•

How about the bispectrum of galaxies?

45

Previous Calculation

•

Scoccimarro, Sefusatti & Zaldarriaga (2004); Sefusatti &

Komatsu (2007)

•

Treated the distribution of galaxies as a continuous

distribution, biased relative to the matter distribution:

•

^{δg = b1δm + (b2/2)(δm)2 + ...}

•

Then, the calculation is straightforward. Schematically:

•

^{<δg3> = (b1)3<δm3> + (b12b2)<δm4> + ...}

Non-linear Bias Bispectrum Non-linear Gravity

Primordial NG ⁴⁶

Previous Calculation

•

We find that this formula captures only a part of the full contributions. In fact, this formula is sub-dominant in the squeezed configuration, and the new terms are dominant.

Non-linear Bias

Non-linear Gravity

Primordial NG

47

Non-linear Gravity

•

For a given k1, vary k2 and k3, with k3≤k2≤k1

•

^F2(k2,k3) vanishes in the squeezed limit, and peaks at the

elongated triangles. ⁴⁹

Non-linear Galaxy Bias

•

There is no F2: less suppression at the squeezed, and less enhancement along the elongated triangles.

•

Still peaks at the equilateral or elongated forms. ⁵⁰

Primordial NG (SK07)

•

Notice the factors of k² in the denominator.

•

This gives the peaks at the squeezed configurations. ⁵¹

New Terms

•

But, it turns our that Sefusatti & Komatsu’s calculation, which is valid only for the continuous field, misses the dominant terms that come from the statistics of

PEAKS.

•

Jeong & Komatsu, arXiv:0904.0497

Donghui Jeong ₅₂

MLB Formula

•

N-point correlation function of peaks is the sum of M- point correlation functions, where M≥N.

Matarrese, Lucchin & Bonometto (1986)

53

Bottom Line

•

The bottom line is:

•

The power spectrum (2-pt function) of peaks is

sensitive to the power spectrum of the underlying mass distribution, and the bispectrum, and the trispectrum,

etc.

•

Truncate the sum at the bispectrum: sensitivity to fNL

•

Dalal et al.; Matarrese&Verde; Slosar et al.;

Afshordi&Tolley

54

Bottom Line

•

The bottom line is:

•

The bispectrum (3-pt function) of peaks is sensitive to the bispectrum of the underlying mass distribution, and the trispectrum, and the quadspectrum, etc.

•

Truncate the sum at the trispectrum: sensitivity to τ_NL (~fNL2) and gNL!

•

This is the new effect that was missing in Sefusatti &

Komatsu (2007).

55

Real-space 3pt Function

•

Plus 5-pt functions, etc... ₅₆

New Bispectrum Formula

•

First: bispectrum of the underlying mass distribution.

•

Second: non-linear bias

•

Third: trispectrum of the underlying mass distribution.⁵⁷

Local Form Trispectrum

•

For general multi-field models, fNL2 can be more generic: often called τNL.

•

Exciting possibility for testing more about inflation! ⁵⁸

Φ =(3/5) ζ

Local Form Trispectrum

k3

k4

k2

k1

g NL

k2

k1

k3

k4

f NL 2 (or τ _NL )

59

Φ =(3/5) ζ

60

Shape Results

•

The primordial non-Gaussianity terms peak at the squeezed triangle.

•

^{fNL and gNL} terms have the same shape dependence:

•

^{For k1=k2=αk3, (fNL} term)~α and (g_NL term)~α

•

^{fNL2 (τNL}) is more sharply peaked at the squeezed:

•

^{(fNL2 term)~α3}

61

Key Question

•

^{Are gNL or τNL} terms important?

62

63

1/k²

64

Summary

•

Non-Gaussianity is a new, powerful probe of physics of the early universe

•

It has a best chance of ruling out all of the single-field inflation models at once.

•

^fNL ~ 2σ at the moment, wait for WMAP 9-year (2011) and Planck (≥2012) for more σ’s (if it’s there!)

•

To convince ourselves of detection, we need to see the

acoustic oscillations, and the same signal in the bispectrum and trispectrum, of both CMB and the large-scale structure

of the universe. ₆₅

Now, let’s pray:

•

May Planck succeed!

66

Now, let’s pray:

• May the signal be there!

67